How do I make a section of hair a different color?
Image above shows eye brow color different from the rest of face.  Single Mesh Shown.  Please excuse the grainy quick render and other shortcuts I took.



Image above shows [material.063] assigned to the [Brow] Particle System using material nodes for Cycles.  Note the multiple Particle Systems of type Hair. For a human there may be reason to consider eyelash hair and eyebrow hair and other locations of hair to be different types of hair.

In a given particle system of type hair in Cycles you assign a material.
First create a UV map for your mesh.
The material can use a texture to control the color directly.
You can paint your personal texture or use one of the standard built in textures.

The hair patches I used in the face image are separate islands.  Your example may be a little different.

In my example I used a texture just to introduce variation to use as input to the color ramp.  This is a common technique for Cycles.

That texture  can be viewed as a gray scale and input to a color ramp.

If a clump of hair has varying but closely related colors this will work fine.

If you want a highly secluded piece of hair you can also created a second or more particle hair systems with another material.  Vertex Groups and Texture can isolate hair systems locations.

You can also create one or more additional well fitting meshes, rather clip on style,  with  different materials.  Particle Edit Mode can place hair precisely.